Seminars and Colloquia by Series

Quantitative convergence analysis of dynamical processes in machine learning

Series
Dissertation Defense
Time
Tuesday, June 25, 2024 - 10:30 for 2 hours
Location
Skiles 006 and online
Speaker
Yuqing WangGeorgia Tech

Zoom link: https://gatech.zoom.us/j/6681416875?pwd=eEc2WEpxeUpCRUFiWXJUM2tPN1MvUT09

This talk focuses on analyzing the quantitative convergence of selected important machine learning processes, from a dynamical perspective, in order to understand and guide machine learning practices. More precisely, it consists of four parts: 1) I will illustrate the effect of large learning rates on optimization dynamics in a specific setup, which often correlates with improved generalization. 2) The theory from part 1 will be extended to a unified mechanism of several implicit biases in optimization, including edge of stability, balancing, and catapult. 3) I will concentrate on diffusion models, which is a concrete and important real-world application, and theoretically demonstrate how to choose its hyperparameters for good performance through the convergence analysis of the full generation process, including optimization and sampling. 4) The generalization performance of different architectures, namely deep residual networks (ResNets) and deep feedforward networks (FFNets), will be discussed.

On Extremal, Algorithmic, and Inferential Problems in Graph Theory

Series
Dissertation Defense
Time
Thursday, May 30, 2024 - 13:00 for 2 hours
Location
Skiles 005 and Online: https://gatech.zoom.us/j/6125656239
Speaker
Abhishek DhawanGeorgia Tech Math

In this dissertation we study a variety of graph-theoretic problems lying at the intersection of mathematics, computer science, and statistics. This work consists of three parts, all of which use probabilistic techniques. 

In Part 1, we consider structurally constrained graphs and hypergraphs. We examine a celebrated conjecture of Alon, Krivelevich, and Sudakov regarding vertex coloring. Our results provide improved bounds in all known cases for which the conjecture holds. We introduce a generalized notion of local sparsity and study the independence and chromatic numbers of graphs satisfying this property. We also consider multipartite hypergraphs, a natural extension of bipartite graphs. We show how certain probabilistic techniques for problems on bipartite graphs can be adapted to multipartite hypergraphs, and are therefore able to extend and generalize a number of results.

In Part 2, we investigate edge coloring from an algorithmic standpoint. We focus on multigraphs of bounded maximum degree, i.e., $\Delta(G) = O(1)$. Following the so-called augmenting subgraph approach, we design deterministic and randomized algorithms using a near-optimal number of colors in the sequential setting as well as in the LOCAL model of distributed computing. Additionally, we study list-edge-coloring for list assignments satisfying certain local constraints, and describe a polynomial-time algorithm to compute such a coloring.

Finally, in Part 3, we explore a number of statistical inference problems in random hypergraph models. Specifically, we consider the statistical-computational gap for finding large independent sets in sparse random hypergraphs, and the computational threshold for the detection of planted dense subhypergraphs (a generalization of the classical planted clique problem). We explore the power and limitations of low-degree polynomial algorithms, a powerful class of algorithms which includes the class of local algorithms as well as approximate message passing and power iteration.

Dynamic Stability in Stochastic Gradient Descent

Series
CDSNS Colloquium
Time
Friday, May 24, 2024 - 15:30 for
Location
Skiles 254
Speaker
Dennis ChemnitzFU Berlin

Please Note: Streaming via Zoom: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Most modern machine learning applications are based on overparameterized neural networks trained by variants of stochastic gradient descent. To explain the performance of these networks from a theoretical perspective (in particular the so-called "implicit bias"), it is necessary to understand the random dynamics of the optimization algorithms. Mathematically this amounts to the study of random dynamical systems with manifolds of equilibria. In this talk, I will give a brief introduction to machine learning theory and explain how almost-sure Lyapunov exponents and moment Lyapunov exponents can be used to characterize the set of possible limit points for stochastic gradient descent.

Validated enclosures of Fourier coefficients in Banach spaces of analytic functions

Series
CDSNS Colloquium
Time
Friday, May 10, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jean-Philippe LessardMcGill University

Please Note: Streaming available via Zoom: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) aimed at establishing the existence of solutions for nonlinear differential equations featuring non-polynomial analytic nonlinearities. Our approach combines the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to effectively construct CAPs. A key highlight is the rigorous management of Fourier coefficients of the nonlinear term Fourier series, achieved through insights from complex analysis and the Discrete Poisson Summation Formula. We demonstrate the effectiveness of our method through two illustrative examples: firstly, proving the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, establishing the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.

This is joint work with Jan Bouwe van den Berg (VU Amsterdam, The Netherlands), Maxime Breden (École Polytechnique, France) and Jason D. Mireles James (Florida Atlantic University, USA)

Thermodynamic formalism for and hitting time statistics for random open dynamical systems

Series
CDSNS Colloquium
Time
Wednesday, May 8, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jason AtnipQueensland University

Please Note: In this talk we present some recent results on thermodynamic formalism for random open dynamical systems. In particular, we poke random holes in the phase space and prove the existence of unique equilibrium states on the set of surviving points as well as find the rate at which mass escapes through these holes. If we consider small holes, through a perturbative approach, we are able to make a connection to extreme value theory and hitting time statistics. Furthermore, we prove a Gumbel's law and show that the distribution of multiple returns to small holes is asymptotically compound Poisson distributed.

Degeneracy of eigenvalues and singular values of parameter dependent matrices

Series
Applied and Computational Mathematics Seminar
Time
Monday, May 6, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/93530218689?pwd=SFkzMXZyZXhZOTdRazhyL1BoVXprdz09
Speaker
Alessandro Pugliese Università degli Studi di Bari Aldo Moro

Speaker will present in person.

Hermitian matrices have real eigenvalues and an orthonormal set of eigenvectors. Do smooth Hermitian matrix valued functions have smooth eigenvalues and eigenvectors? Starting from such question, we will first review known results on the smooth eigenvalue and singular values decompositions of matrices that depend on one or several parameters, and then focus on our contribution, which has been that of devising topological tools to detect and approximate parameters' values where eigenvalues or singular values of a matrix valued function are degenerate (i.e. repeated or zero).

The talk will be based on joint work with Luca Dieci (Georgia Tech) and Alessandra Papini (Univ. of Florence).

The role of symmetry in delay effects on stability

Series
CDSNS Colloquium
Time
Friday, May 3, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
John Ioannis StavroulakisGeorgia Institute of Technology

Please Note: Zoom link for streaming the talk: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

A conjecture of Buchanan and Lillo states that all nontrivial oscillatory solutions of
\begin{equation*}
x'(t)=p(t)x(t-\tau(t)),
\end{equation*}
 with $0\leq p(t)\leq 1,0\leq \tau(t)\leq 2.75+\ln2 \approx 3.44$ tend to a known function $\varpi$, which is antiperiodic:
 \begin{equation*}
 \varpi(t+T/2)\equiv - \varpi(t)
 \end{equation*}
 where $T$ is its minimal period. We discuss recent developments on this question, focusing on the periodic solutions characterizing the threshold case. We consider the case of positive feedback ($0\leq p(t)\leq 1$) with $\sup\tau(t)= 2.75+\ln2$, the well-known $3/2$-criterion corresponding to negative feedback ($0\leq -p(t)\leq 1$) with $\sup\tau(t)=1.5$, as well as higher order equations. 

 We investigate the behavior of the threshold periodic solutions under perturbation together with the symmetry (antiperiodicity) which characterizes them. This problem is set within the broader background of delay effects on stability for autonomous and nonautonomous equations, taking into account the fundamental relation between oscillation speed and dynamics of delay equations. We highlight the crucial role of symmetry in both the intuitions behind this vein of research, as well as the relevant combinatorial-variational problems.
 

Equidistribution and Subconvexity

Series
Number Theory
Time
Wednesday, May 1, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Humphries University of Virginia

A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. If time permits, I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.

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